Classification of Integrable Equations on Quad-Graphs. The Consistency Approach

Abstract:

A classification of discrete integrable systems on quad–graphs, i.e. on surface cell decompositions with quadrilateral faces, is given. The notion of integrability laid in the basis of the classification is the three–dimensional consistency. This property yields, among other features, the existence of the discrete zero curvature representation with a spectral parameter. For all integrable systems of the obtained exhaustive list, the so called three–leg forms are found. This establishes Lagrangian and symplectic structures for these systems, and the connection to discrete systems of the Toda type on arbitrary graphs. Generalizations of these ideas to the three–dimensional integrable systems and to the quantum context are also discussed.

Acknowledgements. This research was partly supported by DFG (Deutsche Forschungsgemeinschaft) in the frame of SFB 288 ``Differential Geometry and Quantum Physics''. V.A. was also supported by the RFBR grant 02-01-00144. He thanks TU Berlin for warm hospitality during the visit when part of this work has been fulfilled.